[Paper Review] The average exponent of elliptic curves modulo $p$

Jie Wu|arXiv (Cornell University)|Jun 26, 2012

Cryptography and Residue Arithmetic12 references4 citations

TL;DR

This paper establishes improved asymptotic formulas for the average exponent $ e_p $ of elliptic curves modulo $ p $, refining prior work by Freiberg and Kurlberg. Under the Generalized Riemann Hypothesis (GRH), it proves $ \frac{1}{\pi(x)}\sum_{p\leq x} e_p = \frac{1}{2}C_E x + O_E(x^{5/6}(\log x)^{4/3}) $, and unconditionally for CM curves, $ \frac{1}{\pi(x)}\sum_{p\leq x} e_p = \frac{1}{2}C_E x + O_E\left(\frac{x}{(\log x)^{1/14}}\right) $, significantly improving error terms.

ABSTRACT

Let $E$ be an elliptic curve defined over ${\mathbb Q}$. For a prime $p$ of good reduction for $E$, denote by $e_p$ the exponent of the reduction of $E$ modulo $p$. Under GRH, we prove that there is a constant $C_E\in (0, 1)$ such that $$ \frac{1}{π(x)} \sum_{p\le x} e_p = 1/2 C_E x + O_E\big(x^{5/6} (\log x)^{4/3}\big) $$ for all $x\ge 2$, where the implied constant depends on $E$ at most. When $E$ has complex multiplication, the same asymptotic formula with a weaker error term $O_E(1/(\log x)^{1/14})$ is established unconditionally. These improve some recent results of Freiberg and Kurlberg.

Motivation & Objective

• To refine the asymptotic estimate for the average exponent $ e_p $ of elliptic curves modulo $ p $, improving upon recent results by Freiberg and Kurlberg.
• To establish a sharper error term under GRH for Dedekind zeta functions of division fields $ L_k = \mathbb{Q}(E[k]) $, reducing the error from $ x^{9/10} $ to $ x^{5/6} $.
• To prove an unconditional asymptotic formula with error $ O_E(x / (\log x)^{1/14}) $ for elliptic curves with complex multiplication (CM), leveraging effective Chebotarev density estimates.
• To unify and simplify the method of Freiberg and Kurlberg using effective analytic number theory tools, particularly in handling sums over $ d_p $ and $ e_p $ via group structure and density theorems.

Proposed method

• The method relies on expressing the sum $ \sum_{p \leq x} e_p $ via the group structure $ E_p(\mathbb{F}_p) \simeq \mathbb{Z}/d_p\mathbb{Z} \oplus \mathbb{Z}/e_p\mathbb{Z} $, where $ e_p $ is the exponent, and relating $ e_p $ to $ d_p $ through $ e_p = \frac{p+1-a_p}{d_p} $, with $ a_p $ the trace of Frobenius.
• The sum is decomposed into two parts: $ S_1 $, summing over $ k \leq y $, where $ k \mid d_p $, and $ S_2 $, over $ k > y $, using dyadic summation and dyadic partitioning to balance error terms.
• For $ S_1 $, the number of primes $ p \leq x $ with $ k \mid d_p $ is estimated using the Chebotarev density theorem applied to the $ k $-division field $ L_k $, with effective bounds under GRH or unconditionally for CM curves.
• For $ S_2 $, the Brun-Titchmarsh inequality and bounds on the number of $ p \leq x $ with $ k^2 \mid p+1-a_p $ and $ k \mid p-1 $ are used, exploiting the fact that $ k \mid d_p $ implies $ k^2 \mid (p+1-a_p) $ and $ k \mid (a_p - 2) $.
• The error terms are optimized by choosing $ y = x^{1/3}(\log x)^{-2/3} $ under GRH and $ y = (C^{-1}N_E^{-2}\log x)^{1/14} $ unconditionally for CM curves, minimizing the combined error from $ S_1 $ and $ S_2 $.
• The constant $ C_E $ is defined as $ \sum_{k=1}^\infty \frac{1}{n_{L_k}} \sum_{dm=k} \frac{\mu(d)}{m} $, which arises from Möbius inversion over the structure of the division fields.

Experimental results

Research questions

• RQ1 What is the precise asymptotic behavior of the average exponent $ e_p $ of elliptic curves modulo $ p $, and how can the error term be improved?
• RQ2 Can the GRH-based error term $ O_E(x^{9/10} (\log x)^{11/5}) $ of Freiberg and Kurlberg be improved, and if so, by how much?
• RQ3 For elliptic curves with complex multiplication, can the asymptotic formula for $ \frac{1}{\pi(x)}\sum_{p\leq x} e_p $ be established unconditionally with a better error term than $ O_E(x \log_3 x / \log_2 x) $?
• RQ4 How does the structure of the $ k $-division fields $ L_k = \mathbb{Q}(E[k]) $ influence the distribution of $ d_p $, and how can this be used to estimate $ e_p $?
• RQ5 What is the optimal balance between the error terms in $ S_1 $ and $ S_2 $, and how does this depend on the choice of the dyadic parameter $ y $?

Key findings

• Under GRH, the average exponent satisfies $ \frac{1}{\pi(x)}\sum_{p\leq x} e_p = \frac{1}{2}C_E x + O_E(x^{5/6}(\log x)^{4/3}) $, improving the previous error term of $ O_E(x^{9/10} (\log x)^{11/5}) $.
• For elliptic curves with complex multiplication, the asymptotic formula holds unconditionally with error $ O_E\left(\frac{x}{(\log x)^{1/14}}\right) $, significantly improving the prior unconditional bound of $ O_E(x \log_3 x / \log_2 x) $.
• The constant $ C_E = \sum_{k=1}^\infty \frac{1}{n_{L_k}} \sum_{dm=k} \frac{\mu(d)}{m} $, where $ n_{L_k} = [L_k : \mathbb{Q}] $, governs the main term and arises from Möbius inversion over the structure of the $ k $-torsion fields.
• The improved error term under GRH results from a refined dyadic decomposition and tighter bounds on the number of primes $ p \leq x $ with $ k \mid d_p $, using effective Chebotarev density estimates with error $ O(x^{1/2} \log(N_E x)) $.
• For CM curves, the unconditional error term is derived using effective bounds on the Chebotarev density theorem with error $ O_E(x^2 \exp\{-B(\log x)^{5/14}\}) $, valid for $ k $ up to $ (C^{-1}N_E^{-2}\log x)^{1/14} $.
• The optimal choice of the dyadic parameter $ y $ is $ x^{1/3}(\log x)^{-2/3} $ under GRH and $ (C^{-1}N_E^{-2}\log x)^{1/14} $ unconditionally for CM curves, minimizing the combined error from $ S_1 $ and $ S_2 $.